1 Relating fracture mechanics and fatigue lifetime prediction Kalman Ziha University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture E-mail: [email protected]Department of Naval Architecture and Ocean Engineering, Ivana Lucica 5, Zagreb, Croatia Abstract: This article presents how to link together the results of fatigue crack growth tests, analytic fracture mechanics and experimental methods of fatigue lifetime predictions. The study at the beginning investigates the effect of mechanical load redistribution among failed and intact micro-structural bonds along fatigue crack growth to final crack at vulnerable locations in materials and structures under cyclic loads. The microstructural load redistribution model is analytically formulated as a mechanical interaction between fatigue crack growth and fatigue endurance on the macroscopic level. The article in continuation investigates how to link the parameters of fatigue crack growth in fracture mechanics to parameters of fatigue life directly from the work done on crack growth determined by testing. The article at the end illustrates the application of the analytic procedure for fatigue lifetime prediction that combines fracture mechanics and the load redistribution model for determination of S-N curve parameters important in structural analysis and design. In this research the fatigue life time parameters are derived from a single fatigue crack growth experiment instead from normally required sets of fatigue tests for different loading conditions. Keywords: fatigue; fracture mechanics; crack growth; interaction; fatigue life; S-N curves; 1. Introduction The aim of this work is to evaluate and verify fatigue characteristics of materials and structures under cyclic loads common to analytic fracture mechanics and experimental fatigue lifetime predictions straightforwardly from precisely recorded Fatigue Crack Growth (FCG) curves. The article keep hold of the characteristics of FCG and FCG rates as defined in fracture mechanics 1–3by using Stress Intensity Factors (SIF) determined through investigation of stress fields in materials at the end of the crack 4-5Fatigue parameters in fracture mechanics in general can be determined analytically, by testing on carefully prepared test specimen 6–7or computationally by using complex numerical models and finite element stress analysis 8-10. The investigations in this article of practical analytical procedures in which fatigue parameters can be evaluated directly from FCG curves are encouraged through the reported improvements in precision of fatigue crack size measurements 1112. The article at the beginning presents the mechanical load redistribution model along the crack to its end on a lattice of microstructural particles in crystalline materials following the concept of crack growth kinetics 13. The study reveals the empirical concept of Cause-and- Effect-Interaction (CEI) 14-17for mathematical formulation of Fatigue crack growth and Endurance Interaction (FEI) induced by overstressing due to load redistribution under cyclic loads. The mechanical interaction model of load redistribution replaces in the article the commonly used numerical methods for fitting of crack growth and crack growth rate 18. Lasting efforts and numerous experiments have been devoted since earlier 19to investigation of fatigue life prediction methods 20and of safer criteria for the prediction of fatigue failures 21. It is recognized earlier in the energy-based approach how the local strain energy density can be taken as a consistent fatigue damage parameter [22- 26]. The strain energy-based life prediction criterion can be extended to include the effects of both mean stress and ratcheting [27]. Moreover, the Neuber’s rule can be interpreted in terms of the total
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1
Relating fracture mechanics and fatigue lifetime prediction
Kalman Ziha University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture
E-mail: [email protected] Department of Naval Architecture and Ocean Engineering, Ivana Lucica 5, Zagreb, Croatia
Abstract: This article presents how to link together the results of fatigue crack growth tests, analytic fracture
mechanics and experimental methods of fatigue lifetime predictions. The study at the beginning investigates the effect
of mechanical load redistribution among failed and intact micro-structural bonds along fatigue crack growth to final
crack at vulnerable locations in materials and structures under cyclic loads. The microstructural load redistribution
model is analytically formulated as a mechanical interaction between fatigue crack growth and fatigue endurance on
the macroscopic level. The article in continuation investigates how to link the parameters of fatigue crack growth in
fracture mechanics to parameters of fatigue life directly from the work done on crack growth determined by testing.
The article at the end illustrates the application of the analytic procedure for fatigue lifetime prediction that combines
fracture mechanics and the load redistribution model for determination of S-N curve parameters important in structural
analysis and design. In this research the fatigue life time parameters are derived from a single fatigue crack growth
experiment instead from normally required sets of fatigue tests for different loading conditions.
0.46 300000 1.048 0.94 2.84 503219 620262 Note: The results in examples are derived in MS-Excel work sheets using intrinsic Solver add-ins with evolutionary
algorithm and Generalized Reduced Gradient (GRG) optimization methods.
7. Discussion
Examples in this article confirmed the agreement of calculated and reported data. However, the differences can be
understood having on mind that the methods of fracture mechanics applied in this research for fatigue lifetime
assessments are developed for steady FCG regime known as ‘region two’, since the experimental methods consider
the whole span of lifetime with high statistical scatter of test data till total failure including the unstable FCG known
as ‘region three’.
13
0
2
4
6
8
10
12
14
16
18
20
0.E+00 1.E+06 2.E+06 3.E+06 4.E+06 5.E+06
Cra
ck le
ngt
h 2
a (
mm
)
Number of cycles (N)
BM TestBM-FEIFSW TestFSW FEI
BM
Nf=
2670
000 N
a=27
2000
0
BMFEI: p=2.17x10-6, i=1.16x10-6
PE: C=2.58x10-10, m=4.66
Ns=
1620
0000
FSWFEI: p=9.47x10-6, i=7.02x10-7
PE: C=3.20x10-10, m=4.66
Ns=
2500
0000
Nf=
4180
000
Na=
4220
000
FSW
22001700
0
500
1000
1500
2000
2500
0.E+00 1.E+06 2.E+06 3.E+06 4.E+06 5.E+06Wo
rk d
on
e o
n c
rack
gro
wth
W(N
)
Number of cycles (N)
BM
Nf=2700000
1700 J cycle/N
(J c
ycle
/N)
2200 J cycle/N
FSW
Figure 5. FCG sizes a(N) for BM and FSW Figure 6. Energy absorption due to work done on FCG
0.0E+00
2.0E-05
4.0E-05
6.0E-05
8.0E-05
1.0E-04
1.2E-04
1.4E-04
1.6E-04
1.8E-04
2.0E-04
0.E+00 1.E+06 2.E+06 3.E+06 4.E+06 5.E+06
Cra
ck g
row
th r
ate
da
/dN
(mm
/cyc
le)
Number of cycles (N)
BMNf=2700000
FSW
1.E-6
1.E-5
1.E-4
1.E-3
1.E-2
1.0 10.0 100.0
Cra
ck (
a)
gro
wth
rat
e d
a/d
N
Stress Intensity Factor
BMBMC=2.58x10-9, m=4.66
K=5.9
K=26(m
m/c
ycle
)
K=8.4
K=53
FSW
FSW
C=3.50x10-10, m=3.78
Figure 7. FCG rates da(N)/dN for BM and FSW Figure 8. Stress intensity factor ranges K(da/dN)
2.50
2.60
2.70
2.80
2.90
3.00
3.10
3.20
3.30
3.40
125 150 175 200 225 250 275 300 325
n
BM
FSW
3.33 for infinite sheet
2.89 for infinite sheet
3.16
2.75
t
(MPa)
100
1000
1E+5 1E+6 1E+7
Stre
ss r
an
ge
(MP
a)
Number of cycles (N)
FSW test
BM test
Failure
Transition
BMA=2.26x10+13, n=3.16
500
400
300
200
BM FSW
FSWA=5.01x10+12, n=2.75
2E+6
600700800900
Figure 9. Inverse slopes n of S-N curves for BM and FSW Figure 10. S-N curves for BM and FSW
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Conclusion
The article elaborates the link between analytic fracture mechanics, fatigue crack growth tests and experimental
lifetime predictions in materials science and engineering.
The results of calculations performed on the reported fatigue test data uphold the thesis revealed at the beginning of
the article that the macroscopically observable fatigue crack growth can be modeled as a mechanical yielding process
induced by overstressing due to redistribution of loads among huge but finite number of failed and intact internal
micro-structural bonds. Interpretation of fatigue experiments in this study allows rethinking of the argument that there
are measurable interactions between the preceding fatigue crack growth and impending fatigue endurance.
Consequently, fatigue crack growth is regarded in the study as an interaction governed by propensity to and intensity
of interaction between the increasing number of load cycles and the resulting crack growth that simultaneously affects
and is affected by the residual fatigue endurance. The fatigue crack growth-endurance-interaction model in the article
replaces the numerical fitting methods for analytic definition of experimental fatigue crack growth and fatigue crack
growth rate for definition of stress intensity factors suitable for steady fatigue crack growth in fracture mechanics.
The interaction model provides the mathematical formulation of energy conservation principle between the work done
on crack growth under external cyclic loadings and the energy absorption due to internal material resistance to fatigue
propagation all over the fatigue life that can be considered as the measure of fatigue strength.
The research of fatigue lifetime in continuation of this article corroborates the thesis that the fatigue life parameters
can be extracted directly from a single experimentally derived FCG curve. The internal energy of resistance to
cracking i.e. the energy absorption equivalent to the work done by cyclic external forces on crack growth can be
calculated through numerical integration of precisely recorded crack size. Accordingly, the article reinterprets the
Basquin’s equation for prediction of fatigue life as the relation of the energy absorption of work done during fatigue
crack growth for different applied cyclic loads until fatigue failure of test specimens or of realistic structures. The
method presented in the article allows assessment of the parameters of S-N curves of importance to life time
predictions directly from a single fatigue crack growth experiment rather than from sets of stress-life tests of lifetime
duration to failure under various loading conditions. The article recommended analytical corrections for the effect of
crack size on crack growth for finite geometries of specimens and structures in order to provide S-N curves for
practical use in their standardized form. The number of repeated physical FCG test to failure can be replaced by a
numerical procedure for different loading conditions illustrated by examples in the article. The method exposed in this
article is found sufficiently simple and numerically accurate for application to problems of fatigue in engineering of
materials appropriate for structural analysis and design.
Acknowledgement
This work was supported by the Ministry of Science, Education and Sports of the Republic of Croatia under
grant No. 120-1201703-1702.
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Nomenclature
a crack size
A intercept of S-N curves
c correction of S-N curve slope for crack size
C intercept of SIF range curves
CE Cause-Effect relation
CEI Cause-Effect-Interaction
E fatigue endurance
fa scaling factor for effects of crack size
FCG Fatigue Crack Growth
FEI Fatigue-Endurance-Interaction
i interaction intensity FEI parameter
m slope of SIF range curves
n inverse slope of S-N curves, also cyclic ratio
N number of loading cycles
p propensity to interaction FEI parameter
PE Paris-Erdogan power rule
R stress ratio
s sensitivity to cracking
S stress range σ or amplitude in S-N approach
SI scatter index
SIF stress intensity factor
Y joint geometric function
W specimen width
K stress intensity factor (SIF) range
σ cyclic stress range
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References
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528-534, 1963.
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the Mechanics and Physics of Solids, 54 1333-1349, 2006.
[3] B. Brighenti, A. Carpinteri, N. Corbari, Damage mechanics and Paris regime in fatigue life assessment of metals.
International Journal of Pressure Vessels and Piping, 104 57-68, 2013.
[4] G.R. Irwin, Analysis of stress and strains near the end of a crack traversing a plate. Journal of Applied Mechanics
24 3 361–364, 1957.
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Rate from Effective Stress Intensity Factor Range. Journal of Testing and Evaluation, 23 3 153-159, 1995.
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